Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspcdvinvd | Structured version Visualization version GIF version |
Description: If something is true for all then it's true for some class. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
rspcdvinvd.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
rspcdvinvd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcdvinvd.3 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
Ref | Expression |
---|---|
rspcdvinvd | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcdvinvd.3 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓) | |
2 | rspcdvinvd.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
3 | rspcdvinvd.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
4 | 2, 3 | rspcdv 3612 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
5 | 1, 4 | mpd 15 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 df-clel 2890 df-ral 3140 |
This theorem is referenced by: imo72b2 40403 |
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